Global well-posedness of the Gross--Pitaevskii and cubic-quintic nonlinear Schr\"odinger equations with non-vanishing boundary conditions

We consider the Gross–Pitaevskii equation on ${\mathbb{R}}^4$ and the cubic-quintic nonlinear Schr"odinger equation (NLS) on ${\mathbb{R}}^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.